Problem 46 - Applied Calculus

Question

For each function: a. Integrate ("by hand") to find the area under the curve between the given \(x\) -values. b. Verify your answer to part (a) by having your calculator graph the function and find the area (using a command like FnInt or \(\int \mathrm{f}(\mathrm{x}) \mathrm{d} \mathrm{x}\) ). $$ f(x)=e^{x} \text { from } x=0 \text { to } x=3 $$

Step-by-Step Solution

Answer

Integral is \(e^3 - 1\), verified by calculator.

Step 1: Write the Integral

To find the area under the curve for the function \(f(x) = e^x\) from \(x = 0\) to \(x = 3\), we set up the definite integral as follows: \[ \int_{0}^{3} e^x \, dx \].

Step 2: Integrate the Function

To integrate \(e^x\), we use the fact that the integral of \(e^x\) with respect to \(x\) is itself \(e^x\). So, \( \int e^x \, dx = e^x + C \), where \(C\) is the constant of integration. For a definite integral, we evaluate at the bounds: \[ \left[ e^x \right]_{0}^{3} = e^3 - e^0 \].

Step 3: Evaluate the Result

Evaluate the expression \(e^3 - e^0\), knowing that \(e^0 = 1\). Thus, the result is \(e^3 - 1\).

Step 4: Use Calculator for Verification

Use a calculator to graph the function \(f(x) = e^x\) from \(x = 0\) to \(x = 3\). On the calculator, use the integral function (like FnInt) to numerically calculate the area under the curve. Ensure that the result approximates \(e^3 - 1\).

Key Concepts

Definite Integral

Integrals can be a bit tricky at first, but once you get to grips with them, they become a powerful tool in calculus. A definite integral is specifically used to find the area under the curve of a graph over a certain interval. This means you are calculating the accumulation of all tiny chunks (or infinitesimally small parts) of area between the function and the x-axis, from one x-value to another.

A definite integral is written as \( \int_{a}^{b} f(x) \, dx \), where \( a \) and \( b \) are the limits of integration, and \( f(x) \) is the function. Unlike indefinite integrals, you don't add a constant \( C \) because the definite integral is a query about the total sum from \( a \) to \( b \).

To evaluate a definite integral, you need two steps:

• Integrate the function (find the integral) without the limits.
• Apply the limits, which involves substituting \( b \) and then \( a \) into the integrated function and subtracting: \( F(b) - F(a) \).

Area Under the Curve

When we talk about the area under the curve in integration, we are referring to calculating the total "weight" or "mass" of an area bounded by a function's graph and the x-axis, within specific limits. This can represent, for example, the total distance traveled, given a speed-time graph.

The area under the curve is crucial for understanding many physical and geometrical problems where direct measurements aren't possible. For example, if you're trying to determine the entire amount of a liquid output described by a time-liquid rate, you'd integrate the rate function over time to get the total quantity.

In the context of the function \( f(x) = e^x \) from \( x = 0 \) to \( x = 3 \), the definite integral \( \int_{0}^{3} e^x \, dx \) helps compute this exact area directly, offering insight into how exponential growth accumulates over time.

Exponential Functions

Exponential functions are a type of mathematical function where the variable is an exponent. You might have encountered the simplest form, \( f(x) = e^x \), where \( e \) is Euler's number, approximately 2.718. This function is unique because its rate of growth is proportional to its value—it grows exponentially.

These functions have real-world applications across various fields, including finance, science, and engineering. For example, they model populations' growth, radioactive decay, and compound interest. Their derivative and integral, both equal to \( e^x \), reflect the function's ongoing proportional growth, making them predictable yet powerful.

The exercise illustrates the use of exponential functions in integral calculus. By integrating \( e^x \) from \( x = 0 \) to \( x = 3 \), we're determining the accumulated growth of \( e^x \) over that period, showcasing exponential functions' dynamic nature in continuous growth scenarios.